Fine structure and Zeeman effect in the b(1)3&u state of...

Fine structure and Zeeman effect in the b(1)311, state of W, by oppler-free polarization spectroscopy

Wen-Haw Jen$ Department ofPhysics, Penn State Ukiversity, University Park, Pennsylvania 16802 Xingbin Xie, b, L. P. Gold, and R. A. Bernheim Department of Chemistry, Penn State University, University Park, Pennsylvtinia I6802

(Received 3 April 1990; accepted 30 August 1990)

The fine structure of the b( 1)311, state of the lithium dimer was observed for the first time and studied using Doppler-free polarization spectroscopy. The triplet state was accessed by excitation of the A ‘Z,*fw= 2,J=~33) --b( 1)311, (u = 9,N= 32,FI ,e) spin-orbit perturbed levels. The remaining two fine structure levels were excited by using the Zeeman interaction to mix levels of different J. Two polarization geometries, rr pump and cr pump, were used to study separately different groups of MJ levels in magnetic fields up to 3 kG. An irreducible tensor derivation of the fine structure and Zeeman Hamiltonian matrix in Hund’s case (b) symmetrized basis functions was carried out and used to determine the perturbation parameters and the spin-spin and spin-rotation fine structure constants by a lit to the Zeeman splitting pattern. The spin-orbit contribution to the fine structure is too small in the high J states to be found accurately. The electronic spin-orbit coupling perturbation for the interaction between the singlet and triplet states was also determined to be Hel(A-b) = 0.1172 & 0.0008 cm-‘.

I. INTRODUCTION

The present work is part of a detailed, comprehensive investigation of the electronic state structure of the lithium dimer. ’ The study of Li, is of considerable interest since there are both similarities and striking differences with the electronic state structure of the hydrogen molecule. Many of the differences are attributable to the presence of an electron core in Liz, while the similarities originate from the mainly a-bonded pair of electrons in the ground state. Both molecules have received considerable theoretical attention, and the recent treatments2” of Li, compare very favorably with the experimental results. However, there have been no attempts, to our knowledge, to treat the Liz triplet state fine structure, while that of H, has been studied both experimen- tally and theoretically.7-23 An experimental study of the Li, triplet fine structure is, therefore, timely and should provoke further tests of the present theoretical understanding of this molecule which is emerging as a prototype for homonuclear diatomics having an electron core under the valence electrons.

The b( 1) 311U state of Lip is of particular interest because of its role in the predissociation of the manifold of states correlating with the 2 ‘S,,, + 2 2P3,2,1,2 atomic states of Li. It also has consequences for alkali dimer lasers.24 Theoreti- cal calculations predict the accidental predissociation of the A ‘Z,+ state through spin-orbit interaction with levels of the b( 1)311, state which are in turn predissociated via interaction with the a( 1>3&+ continuum.25-27 This would consti- tute a loss mechanism in a possible laser application. Experi-

a1 Present address: Department of Chemistry, Wayne State University, De- troit, m 48202.

b’ Present address: Dalian Institute ofchemical Physics, P.O. Box 110, Da- lian, Peoples Republic of China.

mental studies of these effects include lifetime measurements of the different isotopically substituted Li, species28*29 and rotationally resolved spectra of the b( 1) 311U state of 6Li2. 30*31 For comparison, the c 311U (2p7r) state of Hz also exhib- its predissociation, deduced from the variation of lifetimes of the fine structure levels.“*‘9 As with H, , the predissociation in Li, arises from a potential curve crossing with a shallow van der Waals ‘2: state.2*30.3’ In the first observation of the fine structure of the c 311, (2prr) state of Hz ‘** only partially resolved spectral lines could be observed in the Doppler- limited experiments. Complete resolution of the fine structure of several rotational levels was achieved using molecular beam magnetic resonance methods.‘0-‘2,‘9 Subsequently, the fine and hyperfine structure of the similar d 3J&, (3p~) state of H, was studied by electron induced microwave-optical double resonance.*~22

The b ( 1) 311U state of Li, was first observed in collision- induced fluorescence spectra,31 and approximate molecular constants could be derived from an examination of the accidental predissociation of the A ‘2: states.29 In a systematic study of the lifetimes of the A ‘Z,+ levels of the three isotopic molecules, 46 were found to have anomalously short lifetimes due to the accidental predissociation via interaction with the b( 1 )‘IIJ and a( 1)32,+ states28s29 while 2 levels (u=2,J=33andv=3,J=510f6Li2,A’2,+)werefound to have longer lifetimes. These latter are spin-orbit perturbed by certain levels of the b( 1 )311U state which do not interact with the a( 1)32,+ continuum. More precise constants were found with optical-optical double resonance experiments utilizing the A ‘B,+ -b( 1)3111, spin-orbit perturbed levels as intermediate levels.32,33

In the present study the A ‘X,+ (v =~2,J= 33) -b(1)311, (V = 9,N= 32), fine structure states were excited from X ‘2: (v = 0,J = 34 and J = 32) with a single

928 J. Chem. Phys. 94 (2), 15 January 1991 0021-9606/91/020928-14SO3.00 @ 1991 American Institute of Physics

Jeng eta/.: Fine structure of Li, 929

mode cw dye laser. Sub-Doppler techniques are required to resolve the closely spaced A ‘82 and b( 1)311, levels. The spin-orbit selection rules allow only the F, ,e component of the u = 9, N = 32 level of the b( 1)311, state to be excited, but, as discussed below, the other two fine structure sublevels acquire some transition moment in the presence of a magnetic field. The Zeeman interaction preserves the total + / - symmetry but mixes the fine structure components.

L uncoupling can mix the b( 1)311, and a( l)3X,+ states and is responsible for the predissociation. However, this mixing is symmetry forbidden for the above levels, which are not predissociated. The selection rules for the spin-orbit, Zee- man, and L-uncoupling perturbations are summarized in Table I. These are, of course, in addition to the + o$ - and s ++ CI selection rules for total parity and nuclear spin symmetry. The perturbation structure between the A IX,+, b( 1)311,, and a( 1)3E,+ states is also illustrated in Fig. 1.

Initially, attempts were made to use sub-Doppler inter- modulated fluorescence spectroscopy to study the line structure of 6Li2. However, the Zeeman spectra were complicated and unresolvable due to the nearly two hundred Zeeman sublevels of the three fine structure components. These problems were mostly solved by using polarization spectrosco- py34,35 strategies which also improved the signal-to-noise ratio by an order of magnitude. Different polarization arrangements permitted different MJ levels to be observed separately, reducing the complexity of the Zeeman spectra and permitting the fine structure of ‘Li, to be observed and measured for the first time.

II. EXPERIMENTAL

Polarization spectroscopy was first developed and used for Doppler-free studies of electronic transitions in the H atom.“4*35 It has also been used for studies of Na, ,36-3v Cs,, 40*41 Kr,42 SF, ,43 as well as Li, .4M6 The technique has the advantages of high sensitivity, high resolution, and spectral simplification. Molecules in the gas phase are first oriented or aligned by optical pumping with a strong polarized pump beam. The oriented or aligned molecules form an optically anisotropic sample which is both birefringent and dichroic for a weak, polarized probe beam. The polarization charac- teristics of the probe beam are changed as it interacts with the anisotropic sample, and it is this change in polarization of the probe beam which is responsible for the detected signal.

TABLE I. Selection rules for the mixing of energy levels by various terms in the Hamiltonian. Theinteraction categories (a), (b), and (c) are also indicated in Fig. 1.

(a) Spin-orbit mixing

Biii,l,.S,

(b) Zeeman Interaction

-,WH

Cc) L uncoupling

--BN.L

e-e f-f AJ=O AS=O,jtl AA=O,&l A&=0

e-f AJ=O,&l

AM,=0

l?*e f--f

AJ=O AS=0 AN=0

A’P,+(v=2) b’II,(vz 9, N=32) aSI +(N=32) ” J (01 J (b) (cl J

xl::

I

f (-) I fJ(-) s -=F*- a(+la

\

%A 31 F8 et-1 s 33F,- f (+)a

sJ~~~~3’F~-f~t10

FIG. 1. Energy level diagram, symmetry properties, and perturbations re- ferred to in the text and in Table I for the excitation of the A ‘1: -b( l)%,, level from the X ‘S,’ state of ‘Li, .

In polarization spectroscopy the pump beam can either be circularly polarized (producing molecular orientation) or plane polarized (producing molecular alignment). The resulting distributions over MJ sublevels describe the anisotropic distribution of orientations of the angular momentum vector J. In the experiment reported here a linearly polarized beam of radiation is used both to produce molecular alignment and to serve as a detection probe for a Doppler-free transition. As discussed below, this is a convenient choice for studies of the Zeeman effect of the MJ levels.

The experimental arrangement is shown in Fig. 2. The output of a Coherent 599-21 single mode dye laser operating at 14 267.48 or 14 367.94 cm- ’ was used to excite the P( 34) or the R (32) transitions, respectively, to the perturbed levels. The plane-polarized laser output was split into a strong pump beam and a weak probe beam using a microscope slide as a beam splitter. The two beams were focused and counter- propagated through the heat pipe oven, intersecting with a small angle at their focal point. The magnetic field is applied perpendicular to the direction of the two counterpropagating laser beams. The pump beam polarization could be rotat- ed by 90” with a l/2 wave retardation plate.

A mica sheet plus a polarizing prism (P, ) was used to improve probe beam polarization and insure that it was oriented 45“ to both the pump beam polarization and magnetic field before entering the heat pipe cell. After passing through the cell the probe beam is directed through a Babinet-Soleil compensator, an analyzing prism (P, ), a spatial filter, and onto a Hamamatsu 928 photomultiplier tube. The compensator was used to remove some of the ellipticity in the polarized light produced by the cell windows and other optical surfaces. Both the compensator and analyzing prism were adjusted to give maximum extinction of the probe beam while the pump laser beam was blocked. When the frequency of the pump beam corresponds to a molecular excitation, the molecular vapor becomes birefringent, and probe light is transmitted through the analyzer and detected by the photomultiplier.

In the present experiment both the pump and probe beams have the same frequency, being split from the same single-mode laser. As the laser is scanned into resonance with a molecular transition, only those molecules with zero

J. Chem. Phys., Vol. 94, No. 2,15 January 1991

BEAM SPLITTER

DYE LASER

WAVE PLATE

COMPENSATOR

FIG. 2. Experimental schematic for the Zeeman study of the fine structure of the b(I)%, state of ‘Li, by Doppler-free polarization spectroscopy.

velocity longitudinal to the counterpropagating beams will be monitored. A Doppler-free linewidth is, therefore, observed.34

The 95% isotopically pure lithium-6 metal sample, obtained from Oak Ridge National Laboratory, was placed in a 30 cm long x 22 mm i.d. stainless steel heat pipe oven4’ equipped with fused quartz windows tilted at 5” with respect to each other to prevent multiple reflections while also giving nearly equal transmission for light of orthogonal polarization..About 3 Torr of He was admitted to the cell before heating to approximately 1000 K where the spectra were taken. Under these conditions the pressures of Li and Li, are 0.72 and 0.020 Torr, respectively.48 The cell does not oper- ate as a true heat pipe but instead contains a mixture of He and lithium in the central interaction region.

The magnetic field was supplied by a small electromag- net with 3 in. pole pieces and capable of producing fields up to 4 kG. Magnetic field measurement was made with a Hall probe calibrated with the proton magnetic resonance signal measured in liquid H2G. Experiments were conducted for two configurations of the polarization of the pump beam with respect to the applied magnetic field. With E parallel to the H field the resulting spectra consist of r transitions, and with E perpendicular to H the spectra consist of c transitions.

The laser dye used was LDS 722 dissolved in a mixture of propylene carbonate and ethylene glycol. Approximately 40-50 mW of single mode radiation was produced using 4 W of 514.5 nm pump power from an argon ion laser. Wave- length calibration was achieved with the laser-induced I, fluorescence spectrum49 and a confocal Fabry-Perot etalon with 1.28 GHz free spectral range. Spectra were recorded using lock-in detection for both the o-pump and r-pump configurations at different applied magnetic field strengths. All of the measurements reported here were obtained using the P( 34) transition, but the same results could be found using the R (32) transition.

The polarization spectrum in zero magnetic field is shown in Fig. 3 together with the same spectrum taken using fluorescence detection which shows the Doppler broaden-

930 Jeng &al.: Fine structure of Li,

FIG. 3. The polarization spectrum of the spin-orbit perturbed A ‘S,: (u = 2,J= 33)-b( l)‘II,(u = 9,N= 32,F, ,e) levels in zero applied magnetic field.

ing. The central feature in the Doppler-free spectrum is a crossover signa1’0-52 which occurs at. the midpoint of the separation between the two transitions to the mainly singlet and mainly triplet levels. Before presenting the magnetic field dependent results and their interpretation, a brief discussion of the Zeeman effect on the spectral line shapes is necessary as well as a description of the theoretical analysis of the spectrum.

III. ZEEMAN EFFECT IN POLARIZATION SPECTROSCOPY

The &fJ dependence of the signal in polarization spectroscopy has been discussed by a number of authors.3”43,53-55 In order to understand the magnitude of the polarization signal in the presence of a magnetic field one begins with the results for the induced optical anisotropy which are applicable to the conditions that-exist for the present experiment: weak saturation, a geometry with the satu- rating pump beam polarized linearly at 45” to the probe polarization, and with the analyzer prism crossed with respect to the linear incident probe polarization. In this situation the intensity of the probe signal passing through the analyzer is, to lowest order

I=&&-+ (1/4)(Ac~~)~/(l +u2)l, (3.1)

where I, is the probe laser intensity incident on the medium, 6 is a background term due to the imperfect extinction ratio of the two crossed prisms in the probe beam path, Aa = ar (w) - ax (0) is the difference in absorption coefficient with the radiation polarized in the z and x directions 0) is the propagation direction), and L is the length of the medium. The term in (1 + u2) arises from an assumption of a Lorentzian line shape with u= (w - o0 )/F where I’ is the half-width at half-maximum intensity. The Kramers- Kronig relation between dispersion and absorption has also been used. The above expression assumes that residual bire-


Jeng etal.: Fine structure of Li, 931

fringence due to the cell windows has been removed. Experi- mentally, this is largely accomplished with the Babinet-So- leil compensator as described above. It is also found that window birefringence can be reduced by squeezing the windows gently with adjustable clamps.34 If window birefringence is present, or if the analyzing prism is uncrossed, the above expression for the polarization signal contains additional terms.34S35 In the case of small signals the introduction of a finite uncrossing angle has been shown to produce an improvement in signal/noises6 but was not used here.

In the presence of a magnetic field, the energies of the different MJ sublevels of the 6( 1)‘111, state are shifted. For the present study it is important to know the MJ dependence of the polarization signal. The unique M, dependence of the signal arising from the two orientations of E with respect to the magnetic field H makes it possible to selectively detect those different groups of MJ levels that are useful for the spectral analysis, namely those near MJ = & 33 and near M-, =O.

The difference in the absorption coefficient a for light polarized in the z and x directions in the transitions 12) c 11) can be expressed in terms of the anisotropy in the absorption cross section, ~~-a;,, together with the lower level population N.W,J after pumping:

ar -a, =CNMJ(az --a,), JMd

(3.29

where the summation over Mj indicates that all possible transitions between the MJ sublevels of the upper and lower states are being considered. The z and x subscripts which appear in the above equation refer to a space-fixed coordinate system as defined above.

I

In the limit of weak saturation caused by the pump beam and negligible repopulation of the ground state from the excited state, the pump beam depletes the population of the lower Mj level according to”

& NMJ =- W+l [ 1 - ob(J,,J*,~J) Tis3-k)1~ (3.3)

where

N MJ

No

*21 4 Y1,3/2

g

is the total population of the iw, level in the lower state after pumping with the pump beam; is the total population in the lower state before pumping; is the pump laser frequency; is the pump laser intensity; are the lower and upper state MJ relaxation rates, respectively, and is the index of polarization of the pump beam. In the a-pump scheme it is x; in the r-pump scheme it is z. The direction of the applied magnetic field is chosen as the z direction.

The absorption cross section is related to the dipole transition moment operator p by5’

77*21 2

“CJ,,J,) = - 3&“+iC PcJ,J~). (3.4)

The rW, dependence of the dipole moment operator and, consequently, that of the absorption cross section for polarization in the x and z directions can be expressed in terms of Clebsch-Gordan coefficients and a reduced matrix element 5 as follows:54*58

cJ,J+1(1/4)((Jf~J)2+ tJ-MJj2+ (6J+49

-2C[(J+MJ92+3(J+MJ9 -I-2][(J-~Jj2+3(J-~Jj +2]}“2j R transitions CT = XV,*J+f,) ~J,Jtl/4)f[ (J--J)(J+ MJ + lj] “2 + [ (J+ kfJj (J- MJ + 1)]“2}2 Q transitions , (3.5)

~J,J-,(1/4>{(J-iMJi(J--J + 1) + (J+MJ)(J+MJ - 1)

- 2[~J+ic4,9(J--MM,~(J-MJ - lj(J+M, - l)]“‘} P transitions

CJ,J+ I [(J-t 1)’ -M:] R trasitions

0 = 4J,+.M,1J) Zr,JM : Q transitions.

CJ,J- 1 (JZ - MS9 P transitions (3.69

The MJ dependence of the line strength for the pumping schemes for the two magnetic field orientations with respect to the polarized pump beam can be obtained by substituting Eqs. (3.3)-( 3.6) into Eq. (3.2).

Figure 4 shows the plots of the MJ dependence of o, and ox. Figure 5 shows the plots of the MJ line strengths, ]~,(a, -a,)1 and ]a,(~, --crX)]for theP(34) transition. These latter give, quantitatively, the relative MJ dependence of the polarization signal in the u-pump and g-pump schemes. It can be seen from Fig. 5 that the a-pump scheme is most sensitive to the highest IMJ 1 sublevels, and one can assume that the g-pump spectra are mostly due to the MJ = * 33 sublevels. In the r-pump scheme the observed

spectra are mostly due to low ]MJ 1 sublevels as indicated in Fig. 5. As a result of the MJ dependence of the two pumping schemes, the assignment and analysis of the Zeeman spectra is greatly simplified.

IV. THEORY

The evolution of the theoretical treatment of 311 states over the yearsy-2’ has been subject to a number of problems as reviewed by Miller and Freund.l’ In the study reported here the Zeeman interaction must also be incorporated, in- cluding both orbital and spin contributions. The problem, therefore, is one in which the interactions include (a) fine


932 Jeng era/.: Fine structure of Lip

FIG. 4. The M, dependence of the absorption cross section in the z direction, a,, and the M., dependence of the absorption cross section in the x direction, a,.’

structure splitting of the b ( 1) 311u state, (b) perturbation by the A ‘2: state, and (c) Zeeman interaction of both states. Using the irreducible tensor method, expressions for the magnetic field dependence of the A ‘B,f -b( 1)311, perturbed fine structure were determined. In the following discussion only the results of the theoretical derivation are given.

A. Hamiltonian

The total Hamiltonian R’ for the molecule may be written in the limit of the Born-Oppenheimer approximation as

sY=E;:+2Yo +sYr +A?,, (4.1) where Xc contains the kinetic energy of the electrons and

FIG. 5. The M,-line strength, o,? (0; - vx ), for the o-pump schemes, and the M,-line strength, nz (uz - 0; ), for the n--pump scheme.

the Coulombic potential energy due to the interactions of the electrons and nuclei. The terms z” and zr represent the vibrational and rotational contributions, while P’S repre- sents all of the other interactions, such as fine structure, hyperfine structure, and interactions with external fields.

For each of the fine structure components of the v = 9, N = 32 level of the b( 1) ‘III, state of Li,, the first three terms of Eq. (4.1) are identical and can be regarded as a known constant, G!Yc + 2” + .Pr zzz Eb. Also, since the hyperfine structure was not resolved in the spectra, only the fine structure and Zeeman parts of X3 are included. Estimates of the small magnitude of the hyperfine interaction and the possible consequences for the spectral analysis will be presented later. The Hamiltonian becomes

X=E, +SYfs +Zz, where Xfs is the fine structure, and &“, is the Zeeman interaction.

The fine structure Hamiltonian contains interactions of the electron spin with other angular momenta in the molecule and consists of three types of interactions: electron spin-orbit, electron spin-spin, and electron spin-rota- tion9.‘3,14,18.23

A?& = 2Yso + 2Ys, + ;z;p,, . Following current practice,‘9*60 the spin-orbit contribution is expressed as

<zP$, = c iiili’Si al; &TK with iiili = & z -.--L- liK2 (4.2) i r k

where li and si are electron orbital and spin angular momenta, I, is the. orbital angular niomentum of electron i about nucleus K, cr is the fine-structure constant, and Z,,, is the effective nuclear charge.’

The dipolar spin-spin interaction is

ps, I -&& c + si’sj _ 3”y”“‘] [

(4.3) i>j r ,y r/

in conventional notation, and the spin-rotation interaction is

rp,, = yN-S, (4.4) where y is the spin-rotation constant, N is the rotational angular momentum and S = s, + sz .

The Zeeman Hamiltonian Pz, gives the interactions of the components of the molecular magnetic moment with an applied magnetic field.

xz =PB C k,sSi +g,li)*H (4.5) t

The magnetic moment contributions due to nuclear spin and nuclear rotation are ignored.

In summary, the terms of the Hamiltonian in this study are II

R==E, +Zs, +2==s, +c%‘sr +SYz. (4.6)

B. Basis functions

For the N = 32 level of the b( 1)311, state of “Li,, the rotational constant is B = 0.8342 cm ~- ’ and the spin-orbit

J. Chem. Phys., Vol. 94, No. 2.15 January 1991

Jew eta/: Fine structure of Liz 933

constant A is of the order of 0.1 cm- ‘.32*33 For the high J values encountered here the condition of [AAl 4 BJ( 2s) 1’2 is satisfied and Hund’s case (b) is the most appropriate scheme to describe this level.59

The wave functions are designated59*6’ as 1nA.v) ISNAJM,) h w ere the first A characterizes the electronic angular momentum of the state, and the second A is its component along the internuclear axis; InAv) is the electronic and vibrational wave function while [SNAJMJ) is the coupled case (b) wave function. The three fine structure components of the b( 1)311T, state are El (J = N + l), F, (J = N) , and F3 (J = N - 1) . In’ the presence of a strong magnetic field Jis no longer a good quantum number, and N and S decouple from each other to precess around the field direction, separately. The basis function can be decoupled via the linear transformation62

jn,A,u> ISNAJM,)

= InAv) 2 (SN~~hl,lJM,)IN~~)ISM~>, MNMS

(4.7) where the (SNI)/I,M, I Jlw,) Clebsch-Gordan coefficients are, for this work, expressed in terms of the Wigner 3-jcoefficients:

WQKMvIJM,)

= (- 1)S-N+IW”(2J+ 1)“2 J

-M . 3 >

(4.8) Because ‘Li nuclei are bosons (I = 1 ), the total molecu-

lar wave function must be symmetric with respect to inter- change of the nuclei. Moreover, the total nuclear spin for the “Li, molecules studied in this work is T = 2,0 (o&o) with symmetric spin functions. This means that the X ‘Xlp+ (J= 34) level has “ + ” total parity, and the upper state A ‘xc,+ -b( 1)311, levels observed have “ - ” total parity. These are shown in Fig. 1 together with the appropriate e/f parity and the (s), (a) symmetry designations. The b( 1) 311U basis functions in case (b) must also be properly symmetrized giving

xk ( - UNln, - A,v)IS,N, - WJ’Jf,)~ (4.9) for the total molecular wave function with the negative linear combination for the orfho moleculars with T = 2,0, and the positive linear combination for the para molecules with T = 1. Table I summarizes the selection rules indicated by arrows in Fig. 1 for the mixing of energy levels by off-diagonal terms in the matrix of the Hamiltonian which for this situation are (a) the spin-orbit interaction, (b) the Zeeman interaction, and (c) the L uncoupling. Since the L uncoupling cannot mix states of different total parity or different nuclear spin symmetry, predissociation by LZ ‘Zt (N = 32) does not occur, which permits the observation of the sharp spectra in this study.

C. Matrix elements Based on second order perturbation theory, the shift of

energy levels due to fine structure matrix elements off-diag-

onal in N. is proportional to [&s UW’)]2/ [E(N) - E(N’)].Thematrixelements [H, (N,N’)] ofthe b( 1)311, state of the 6Li, are estimated to be of the order of 0.1 cm - ’ or less, and the energy separation between adjacent N levels for N = 32 is about33 50 cm _ ‘. The shift due to perturbation by adjacent N levels is, therefore, about (0.01/50) cm- ’ -6 MHz or less. Since this value is much smaller than the experimental uncertainty ( -45 MHz), fine structure matrix elements off-diagonal in Nare neglected, and only the N = 32 matrix elements are included in our analysis. The spin-orbit matrix element between the A ‘8: (u=9,J=33) and b(1)311, (u=9,N=32,Fl,e) levels is taken as a parameter in the Hamiltonian to be fitted.

The problem of the fine structure of the c 311, (2pr) state of Hz following a case (b) coupling scheme has been studied by many authors.12-18 The b( 1 )311, state -of Li, is similar to the c 311, (2pr) state situation in H,, and we draw upon this previous experience following the irreducible tensor method.62-67

The spin-orbit Hamiltonian in Eq. (4.2) can be expressed as the scalar product of the irreducible tensor operators T(Zi) and T(si) as

SFso = C d,T’(Z,)-T’(q). (4.10) I

After N and S are decoupled from J, and s1 and s2 are decoupled from S, the reduced matrix elements of the single electron operators 2 T ’ ( Zi ) and T ’ (ii ) are found and trans- formed to molecular coordinates. Combined, they give the spin-orbit matrix element diagonal in all quantum numbers as

(SNAJ IPs,, ISNAJ)

=(-1) s+J+l-“$-[~ ; ;3(2N+ 1)

(4.11)

where the Wigner 6-jand 3-jcoefficients have been used. The quantity a, is the one-electron orbital integral

a, = (n-\+-) (4,12) usually called the molecular spin-orbit parameter. It is dis- tinct from the spin-orbit coupling constant A which for the CG- configuration is ii(311,,a~) = a,/2. An analytical expression for the zs, matrix element is

(SNAJ j2Yqo ISNAJ) = U2N.S a, A2 NW+ 1) ’

(4.13) P

where

N-S = (1/2)[J(J+ 1) - N(N+ 1) - S(S+ l,].

The spin-orbit matrix elements diagonal in N for the fine structure components are


934 Jeng eta/.: Fine structure of Li,

(F,,J=N+ lIX~,IF,,J=N+ 1) + ,;;;, )

(4.14)

(F2,J=NIXs,[F2,J=N) = -g N(;;zl) )

(4.15) and

(F3,J=N- llXs;IF3,J=N- 1) = -is.

(4.16) The N dependence in the denominators suggest a decreasing contribution of spin-orbit interaction to the fine structure with increasing N. In the present case of N = 32, the spin- orbit interaction is found to be very small compared with the other interactions and cannot be fit accurately.

The magnetic interaction between two electron spins in Eq. (4.3) can be written as a scalar product of two spherical tensors of the second rank6a

C2(@) ps, = - &P;d~2(s* % 1.3,

r 12 (4.17)

where t9 and 4 are the polar and azimuthal angles of r,2 with respect to the fixed coordinate system, and C * ( 64) are Ra- cab’s normalized spherical harmonics. Again, the evalua- tion of the matrix elements proceeds by uncoupling the angular momenta to permit determination of one-electron reduced matrix elements. The final result is

(S’N’A’J’M; 12Ys ISNAJiw,)

z-&,,6,;,,( _ l)J’+N’+N-- >

EWV’ + 1)(2N+ 111’”

[( N’ 2 N x 1 0 -1 > BO

f1 B, , > 1 (4.18)

where the Wigner 9-j coefficient was found to have the value 0.1111, and the “ -+ ” sign is the same as the linear combination in Rq. (4.9) which is negative in the present case.

Since only one N level was observed in this study, the experimental data could not distinguish the B, and B, constants from each other. The quantity in the square bracket of Eq. (4.18) is fitted as the spin-spin constant and defined as A.

The spin-rotation interaction N-S, can be evaluated by the irreducible tensor method or by applying angular momentum algebra. Since both N and S are good quantum numbers in the case (b) coupling scheme, the calculation is simple. The expressions for the spin-rotation interaction of the three fine structure components are

(F,,J=N+ lI&“,,IF,,J=N+ 1) = NY,

(F,,J= NI*~,IF,,J= N) = -“/t (F,,J=N- lI&+“>,.IF,,J= N- 1) = - (N+ l>r.

(4.19) Table II shows the zero-field Hamiltonian matrix in the case (b) basis set.

The interaction of the molecules with the external magnetic field H can be described by spherical tensor algebra as

2, =~a C [gsT’(s;)*T’W) +gJ’(WI?H)]. I

(4.20) If the direction of the magnetic field is defined as the z axis, the scalar products of above equation are reduced to one component,

>yz =/h&C [&m~i) +Gwi)]. (4.21) I

and from the Wigner-Eckart theorem, the Zeeman matrix element becomes

=~~$++JL~H( - l)--+ J’ 1 J -M; 0 iw, >

x C(g, (S’N’A’J’I]T’(.q) IISNAJ) I

+g,(S’N’A’J’IIT’(Zi)llSNAJ>,. (4.22)

After evaluating the reduced matrix elements the Zeeman interaction becomes

TABLE II. The zero-field Hamiltonian matrix elements in the case (b) basis set.

A ‘2; (J= 33) EAP HAbb 0 0 b”n,(F,,N=32) HA6 Eb + 0.015 15a,.+ 32y+ 3.84471’ 0 0 b%-I,(F,,N=32) 0 0 E,, - 0.000 473 5a, - y - 8.0632/2. = 0 b%,([,,N= 32) 0 0 0 Eb - 0.015 62a, - 33~ + 4.2185/1’

‘E, = the deperturbed position of the A ‘2: (II = 2,J = 33) level. bHAb = spin-orbit matrix element between the A ‘2: (u = 2,J= 33) and b “II, (IJ = 9,N= 32,F, ,e) levels. =Eb = H, + H, + H, of the b ‘II, (0 = 9,N= 32) level. arr = molecular spin-orbit parameter. y = spin-rotation constant.

/I= 32 2 32 = 1 0 -1 B, spin-spin constant.


Jeng etal.: Fine structure of Li, 935

(S’N’AJ’M;j&“,ISNAJM~) = 6,,,p,J( - l)J’-MJ :, “M > C-1) N+S+J+‘[(2J’+ l)(WS I)]‘”

J

x[g$ “s’ Jx&+&(y’ “N’ s) X(--l) N--h+1[(2N’+ 1)(2N+ 1,1”2~’ ; “,)A]. (4.23)

No matrix elements exist between states of different MJ. There are nonzero diagonal as well as off-diagonal Zee-

man matrix elements in J, and their analytical expressions are proportional to MJ and to the magnetic field in the z direction, H:

(JIXiIJbWH,

(JIXzlJ+ I>- (J+ 1)2 -M: “2

I H.

(Jr+ 1)’ The off-diagonal matrix elements mix the fine structure

components with AJ = & 1. Because of this Zeeman mixing, all of the fine structure components become accessible to excitation from the ground X ‘8: level.

One of the results that emerges from inspection of the Zeeman matrix elements is the MJ dependence of the mixing of the fine-structure levels. For the highest values of ]MJ I, there is no mixing. This means that for the u-pump scheme, which is most sensitive for MJ = f 33, only the F, component will be observed through its spin-orbit mixing with the A ‘2: state. In this case the matrix of the Hamiltonian will be reduced to a two by two for fitting the a-pump spectra. For the low values of IM, I, F, , F2, and F3 are all mixed and, therefore, will all be observed in the s--pump scheme which is most sensitive to low IMJ I.

V. RESULTS AND ANALYSIS A. The u-pump spectra

As explained previously, the spectra obtained in the U- pump scheme are mostly due to the MJ = f 33 sublevels, and the matrix of the Hamiltonian can be reduced to a 2 by 2 matrix with only the A ‘Hz (v = 2,J== 33) and b 31T, (u = 9,N = 32,F, ,e) basis functions included. Examples of the a-pump spectra in various magnetic fields are shown in Fig. 6. Crossover resonances appear, the two central peaks at 85 G being an example, and are designated with an “x” in the figure.

After the M$ quantum numbers for the peaks were assigned, the a-pump spectra were fitted to the 2 by 2 Hamilto- nian withMJ = - 33 and -I- 33, respectively. Figure 7 gives the experimental data and the fitted MJ f 33 transitions at various magnetic fields. The crossover signals have not been included in Fig. 7. The excellent fit is characterized by a variance” of 0.49, indicating that the deviation between the observed and calculated data is less than the experimental uncertainty. The average experimental uncertainty of the U- pump spectra is about 45 MHz. From this fit the deperturbed separation between the A ‘B,f (v = 2,J=== 33) and the b 311U (v = 9,N = 32,F, ,e) levels, was obtained as well as the spin-orbit matrix element between these two states.

t The fine structure constants of the b( 1)311, state could not be determined from the a-pump spectra alone due to its in- sensitivity to the other two fine structure components.

The experimental data and the correlation matrix as well as a detailed discussion of the spectra for the a-pump fit are given in the PAPS document available from the AIP.58

8. The z-pump spectra

In order to observe the remaining fine structure levels of the b( 1) ‘IIU state, the mixing caused by the Zeeman interaction is used to access all of the fine structure components in the r-pump scheme as shown in Fig. 1. The amount of singlet character in each MJ level varies as a function of the extent of Zeeman splitting. It also is different for each fine structure component. Since the observed intensity is the con- volution of the square of the singlet character with the MJ line strength, the position of maximum intensity in the experimental r-pump spectra at different magnetic fields may not always be identical to MJ = 0 or even to the same value of MJ. This introduces an error if the fit is attempted assum- ing that the maximum spectral intensity corresponds to MJ = 0 for all measurements.

Because more Zeeman levels contribute to the observed spectra in the n--pump scheme, and since the assignments of M, contain some uncertainty, the analysis of the r-pump spectra requires more effort. In fitting the r-pump data, the values of the deperturbed position of the A ‘8: level and the spin-orbit matrix element determined from the results of the c-pump fit were held constant. The peak positions of each transition were then fitted by the 4 by 4 Hamiltonian which includes the fine structure interactions.

In the first attempt to fit the r-pump data, MJ was set equal to 0 since the MJ = 0 sublevel possesses the highest line strength [Fig. 5 (b) 1. The assigned experimental uncertainty of each line was then set according to its linewidth with weaker, broader lines having larger deviations. After the first fit, a set of the fine structure constants was obtained which was then used to calculate the intensities and eigenvalues of all 67 MJ sublevels at all experimental field values, where the intensity is a result of the product of the square of the singlet character (of each sublevel) with the MJ line strength. The calculated line position of the MJ sublevel having the maximum intensity corresponds to the observed peak position. These Iw, values usually differed from positions of MJ = 0 in the low field region. For the best fit, i.e., that with the smallest variance, there remained some deviation between the calculated positions of Mj = 0 and the experimental peak maxima at low fields due to the above variation in the MJ value for the observed peak maxima.

J. Ctiem. Phys., Vol. 94, No. 2,15 January 1991

936 Jeng &a/.: Fine structure of Li,

MJ'+%

T kdJ=-33

‘3

-I 0 I 2 GHr

MJ =+33 +j, MJ=-33

‘3-j(

+33 -v ‘33433

;I:- A_““-

I 2 3Gz ---Y---f++-w

'T, "I+--- -I 2 GHz

FIG. 6. The c-pump spectra of the Zeeman splitting in various magnetic fields. The numbered features in the spectra correspond to the numbered branches in the Zeeman pattern in Fig. 7. The Mi and crossover assignments are also labeled.

In the second fit, the experimental deviations of each data point were taken as the difference between the calculated eigenvalues of maximum spectral intensity and that of the MJ = 0 sublevel. This deviation reflects the correctness of the assignment of Iw, = 0 in the Hamiltonian. With these adjusted deviations, another set of constants was obtained.

In the third and subsequent fits the deviations were adjusted again using the same procedures as in the second fit. The fitting process was then stopped when a converged result was obtained, i.e., the set of fine structure constants obtained were within the deviations of the previous set.

The complexity of the n--pump spectra is shown in Fig. 8. In the field regions between 218 and 780 G the line broadening is very evident and is likely due to the participation of many Zeeman sublevels as explained above. A Zeeman plot

of just the calculated MJ = 0 eigenvalues vs experimental data for the final fit is presented in Fig. 9 where it should be recognized that the MJ = 0 positions are not equivalent to the peak positions at low magnetic fields. The numbered branches in the Zeeman plot correspond to the numbered features in the spectra of Fig. 8. There are supposed to be four branches, three of which are mostly triplet fine structure components, and one which is mostly singlet. In the high field limit, the branches corresponding to MS = + 1 and - 1 in the decoupled representations were tuned too far away from the singlet level to have any transition intensity and could not be observed. In addition, the transitions in branch 1, which corresponds to M, = + 1, were too weak to be observed even in the low field region. The reason for this becomes evident from a calculation of the spectral intensity.


Jew eta/: Fine structure of Li, 937

FIG. 7. The a-pump Zeeman splitting pattern and the fitted M, = 0 eigenvalues. Crosses are experimental data, and the circular points are fitted eigenvalues.

Intensity and eigenvalue calculations that are included in the supplementary PAPS document58 also indicate that branch 3 will have a large experimental uncertainty at fields lower than 700 G. This explains the systematic deviation of MJ = 0 in this branch in Fig. 9, where the experimental line positions lie lower in energy than the calculated ones for MJ = 0. The three peaks at fields higher than 1.5 kG in Fig. 8 are the spin-orbit perturbed singlet and triplet levels with M, = 0, and their crossover signal. Details of the assignments of the r-pump spectra, intensity and eigenvalue calculations, and results of different fits are given in the PAPS document5’ where it is also shown, that some of the r-pump features are actually due to high ]MJ 1 sublevels.

Using these considerations, the variance of the best r- pump fit is 2.06 which is not as satisfactory as the a-pump fit where the variance was 0.49. The r-pump fit also can be affected by several factors other than the &iJ assignment of the observed peak positions. One of these is the broadening of the Zeeman components of the IV = 32 level by the hyperfine structure interaction which was not included in the Hamiltonian. The levels under study here have an even nuclear spin symmetry with total nuclear spin T = 2,0 (ortho) . However, the hyperfine structure is expected to be small, as shown below. The fit could probably be improved with an analysis in which the MJ value of the spectral peak is used as a least squares variable. However, this approach is a complicated one. Experiments on other b( 1) ‘H,, levels would be a more desirable approach for obtaining improvements on the fine structure constants.

Recent work on Na, has shown that the hyperfine structure in 311 and 3A states is dominated by the Fermi contact interaction between the nuclei and the unpaired electron in the o nsvalence orbital.6Y,70 Moreover, the magnitude of the interiction can be approximated from the isotropic hyperfine constant measured in the ‘S,, atomic ground state. The measured7’ hyperfine splitting of 228.205 26 MHz for the

6Li 2 z&,2 ground state yields an isotropic hyperfine coupling constant b[2 ‘S, ‘Li] = 151.1 MHz and b[b( 1)311,, 6Li, ] = 38.0 MHz if the same primitive LCAO arguments hold for b( 1)311,, ‘Li, as in Na,. This value is five times smaller than the corresponding value observed in Na, triplet spectra, and explains why no readily recognizable hyperfine features were observed in the b( 1) 311u, ‘Li, experiments reported here. It is also a value that is comparable to the stan- dard deviation in the spectral fit. It is possible that experiments on ‘Li, could yield a hyperfine interaction due to the nearly four times larger expected splitting. However, appropriate singlet-triplet intercombination transitions would have to be found.

C. The perturbation parameters and fine structure constants

The spin-orbit matrix element between the A ‘2: and b ( 1) 311u electronic states is HAb, = 653.3 f. 4.6 MHz as determined from the o-pump experiments. The electronic por- tion of the spin-orbit matrix element is obtained by dividing the observed HAb by the calculated ] (u = 21~ = 9) ] = 0.38 vibrational overlap and the I (3II,,,J = 331311,F, ,e,N= 32) I = 0.489 case (a&case- (b) rota-

tional overlap factor3” to yield

IHel(A-B)I = 3516 +25 MHz = 0.1172 &to.0008 cm.-‘. (5.1)

Table III shows the excellent agreement for the comparison of the electronic A-b spin-orbit perturbation factor and deperturbed separation between the A ‘Z,t and b( 1) 311, levels obtained from this Zeeman study with those obtained by Xie and Field32 using line shape analysis. The accuracy of the parameters was improved by an order of magnitude in this study. The spin-orbit interaction between A ‘B,+ and b( 1) ‘III, is smaller than that observed in Na, , and a comparison between the values in Na, and Li, has already been discussed.32

For the fine structure splitting, the molecular spin-orbit parameter, urr defined in Eq. (4.12) could not be extracted from the experimental r-pump data. This is not surprising since the spin-orbit interaction in Li, is very small, and its contribution to the fine structure splitting in this work is further suppressed by a factor of l/N with Jv= 32. How- ever, an estimate of the fine structure spin-orbit constant arr can be made from the spin-orbit matrix element between the A ‘22 and b 3Hu states. Both of these quantities are related to the atomic spin-orbit parameter 5 in the single configuration approximation.59 For arr the relation is

4 = f (5.2)

if the atomic overlap integral S,, = (prA ]prn) for the rr bond in the A-B diatomic molecule is neglected.

The spin-orbit Hamiltonian of Eq. (4.2) can also be written


938

MJ=-33

4a X

Jeng et al.: Fine structure of Li,

MJ=-33

?,T lg x x

418G

I -I 0 I 2 GHz

MJ x-33

41 x x

-I o- I 2GHz

M~?+33?

f+-dYT X x X?

608 G

M~=+33

XM

n

-I 0 I 2GHz

FIG. 8. The or-pump spectra ofthe Zeeman splitting in various magnetic fields. The numbered features in the spectra correspond to the numbered branches in the Zeeman pattern in Fig. 9. The M, and crossover assignments are also labeled.

GHr 1.5

,._ rFs, = c U,l,~S,

i

- =p2i[z~zsiz + (~/2)(Z;tS, + I,$+ ,I. (5.3)

% %+ 3 FtFi-F$i+tF +t ++ ++

0.0 - *-

I ‘I +k c "t4

-0.5 . +

b 11 I I I I 0 1000 2000 3000 G

FIG. 9. The r-pump Zeeman splitting pattern and the fitted MJ = 0 eigenvalues. Crosses are experimental data, and the circular points are fitted eigenvalues of MJ = 0.

The spin-orbit matrix element between the A ‘2: and b( 1)311, states arises from the 1/2(Zi+s; + liPsi+ ) terms of&4,, and can be cakulated using either the ( l/2)1 i+ s; or ( 1/2)Zi-s,+ part.59 For example,

TABLE III. The electronic A ‘2,: -b ‘lIU spin-orbit perturbation factor, H” (A-b), and the deperturbed separation, AE' =E[A'Z,t(u=r2,J=33)] -E[b”n,,~u=9,N=32,F,,e)].

He’ (A-b) (cm-‘) AE' (cm-‘)

This work 0.1172 + 0.0008 - 0.0244 j, 0.0004 Ref. 32 0.114 + 0.006 - 0.020 rfro.001


Hadb = (A ‘2: (u = 2,J= 33) 1

X(1/2)6 &1,-s*+ lb 311,(u = 9,N= 32,F,,e)). I

(5.4) It is easier to evaluate the matrix element of Eq. (5.4) in the case (a) basis se&l2 and the selection rule of AR = 0 must hold. Eq. (5.4) can be expressed as

H.db = (U = 21~ = 9)(3111,,,J= 33j311,F, ,e,N== 32)

x bp?Jm = 0) 1

X(1/2)CiiiliS:la~~+B(n=O)),

H=’ = (cpo,~~~ = 0) I

X(1/2)~a,r~s~~a~~+(n=o,), 1

= (1/2)(alU -I?$ = (1/2)a-, (5.5)

where the electron-spin wave function for the A ‘8& state is

IA = 0,8 = 0,!2= 0) = IcTgaa,~).

and for the b ‘II, substate is

IA = 1,8 = - l,fl= 0) = Iu$r+~).

The quantity a - is defined as

a -- = (alail - IT+ ) and a ‘- is related to the atomic spin-orbit parameter [ by

a- = Jz[. (5.6) Combining Eqs. (5.2), (5.5), and (5.6), the approximate molecular spin-orbit parameter is

a, = @H” = 4972 f 35 MHz. (5.7) The fit of the r-pump spectra was improved using this approximate value held as a constant in the Hamiltonian. However, the sign of arr was not certain. The variance of the best fit was 2.13 with a, = - 4972 MHz and 2.06 with arr = + 4972 MHz. The variance did not change very much with the different signs because the fit is not very sensitive to as as mentioned previously.

If it were possible to make a meaningful comparison of the fine structure spin-spin interaction in Li, with that in H2, Eq. (4.18) for the spin-spin matrix element would be cast into the analytical form of

H -E 3(N+3)(2N*S + 1) - 4N(N+ 1) ss - 2(2N- l)N(N+ 1)(2N+ 3)

x [3 -N(N+ l>lBo I

9_ ( - 1)” @W+ W,J , (5.8)

with the f sign originating from the appropriate linear combination of functions in Eq. (4.9). If data for another rotationallevel of b( 1)311T, of 6Li, becomes available, values

TABLE IV. Fine structure constants for the b( l)‘II,, (u = 9,N = 32) state of ‘Li, .

Molecular spin-orbit parameter a, = 4972 & 35 MHz” Spin-rotation constant y=8.4+1.1MHz

Jew efal.: Fine structure of Li, 939

il Approximated from HAI and equal to 2A where A is the spin-orbit coupling constant.

of B, and B, can be obtained. Table IV summarizes the fine structure constants.

As yet, there are no theoretical treatments of the Li, b( 1)3111, fine structure. Since the current comparisons between theory and experiment for the other molecular constants of Li, is so satisfactory, such a comparison for the triplet state fine structure would be very interesting.

The deperturbed fine structure splittings of the u = 9, N = 32 of b 311, state of 6Li, are Fz - F, = 267 f 3 1 MHz, and F, - F3 = 719 rfi 73 MHz. The relative order in decreasing energy of these levels is Fz , F, , F3 which is the same order found in the c 311U (2p7r) and d 311U (3p7r) states of H, .‘“,73 At the beginning of the r-pump fit the energy order of the fine structure components was not known, and the best fit was not obtained until the above ordering was found.

VI. SUMMARY

Through spin-orbit perturbation with the A ‘8,+ (u = 2,J = 33) level, ‘Li, molecules were directly excited into the b( 1)311, (u = 9,N = 32,F, ,e) level from the ground X ‘2: state. The remaining two triplet fine structure levels F2 and FI, were made accessible to excitation from the X ‘“z state by the Zeeman mixing in a magnetic field. Ex- periments were performed in magnetic fields up to about 3 kG.

The spectra were simplified by the techniques of polarization spectroscopy. In the o-pump configuration, the polarization of the pump beam is oriented 90 deg with respect to the magnetic field direction, and the experiment is most sensitive to highest IMJ 1 sublevels. In the r-pump configuration, polarization of the pump beam is oriented parallel to the magnetic field giving maximum sensitivity to low IMJ I levels.

The fine structure and Zeeman Hamiltonian matrix elements were evaluated in symmetrized Hund case (b) basis functions by the irreducible tensor method. Analytical Iw, dependence of the Zeeman matrix elements off-diagonal in J indicate that there is no fine structure mixing for the n/r, = + 33 sublevels. As a result, only the F, component of the triplet manifold was observed in the g-pump scheme. From the o-pump data the deperturbed separation between the A ‘8: (u = 2,J= 33) and 6(1)3111,(u=9,N= 32,F,, e) levels and the spin-orbit matrix element between these two states were obtained.


940 Jeng eta/.: Fine structure of Li,

t F2 267t31 MHz

---~ ------ +”

733 f 12 MHz 719 f 73 MHz

l- F3

FIG. 10. The deperturbed separation between A ‘2: (v= 22,J= 33) and b(l)“lI,,(u=9,N=32,F,,e) and the deperturbed fine structure splitting.

AT: b(l)3Il,

In the r-pump scheme a number of low IVY sublevels of each of the three fine structure components were observed which required a more detailed procedure for analysis. The spin-spin and spin-rotation contributions to the fine structure constants of the b( 1) 311, state were extracted from the r-pump spectra. Due to the large value of N, the contribution of the spin-orbit interaction to the fine structure splitting was much smaller than that of the other two interactions and could not be obtained directly. An estimated value was extracted from the electronic spin-orbit matrix elements between the A ‘2: and b( 1 )311, states.

The spin-orbit deperturbed separation between the A18,+(u=2,J=33) and b(1)311, (u=9,N==32,FI,e) levels and the deperturbed fine structure splittings of the b( 1)311, level are summarized in Fig. 10. The spin-orbit matrix element between A ‘2: (u = 2J= 33) and b 311u (u = 9,N = 32,F, ,e) is 653.3 f 4.6 MHz, and the electronic spin-orbit matrix element between these two states is 35 16 + 25 MHz or 0.1172 f 0.0008 cm - ‘. The estimated molecular spin-orbit parameter of the b( 1) 311u N f 32 level is a, = 4972 k-35 MHz, the spin-spin constant is il = - 51.2 a 2.4 MHz, and the spin-rotation constant is y= 8.4+ 1.1 MHz.

Ultimately, it would be desireable to make a comparison of the fine structure parameters in Li, with those in H, . Both molecules have received detailed theoretical attention and serve as testing grounds for theory in comparisons with~experiment. The presence of core electrons in Liz not only makes these investigations challenging, but particularly rel- evant since most molecular species contain electron cores. However, a meaningful comparison of the ‘IIp fine structure in Li, with that in H, will have to wait until at least one more rotational state in Li, is studied. It will then be possible to separate the contributions to the spin-spin interaction. Also, a level with low N would give an accurate value of the spin- orbit interaction for comparison with the already existing low N results in H,. Although some values for the spin- rotation constant exist for H2,74 experiments at high N would be desireable to obtain a precise value.

ACKNOWLEDGMENTS Support from the National Science Foundation and the

Donors of the Petroleum Research Fund of the American Chemical Society is gratefully acknowledged. One of us, (W.H.J.) would like to express her appreciation to Professor R. W. Field and his research group for their hospitality and help during her stay with them.

’ R. A. Bernheim, L. P. Gold, P. B. Kelly, C. Kittrell, and D. K. Veirs, Phys. Rev. Lett. 43, 123 (1979); R. A. Bernheim, L. P. Gold, P. B. Kelly, T. Tipton, and D. K. Veirs, J. Chem. Phys. 74,2749 (1981); R. A. Bern- heim, L. P. Gold, and T. Tipton, ibid. 78, 3635 (1983); R. A. Bernheim, L. P. Gold, C. A. Tomczyk, and C. R. Vidal, ibid. 87,861 (1987).

‘M L. Olson and D. D. Konowalow, Chem. Phys. 21,393 (1977); 22,29 (1977); D. D. Konowalow and M. L. Olson, J. Chem. Phys. 67, 590 (1977); 71,450 (1979).

‘D. D. Konowalow and P. S. Julienne, J..Chem. Phys. 72,5815 (1980). ‘D. D. Konowalow and J. L. Fish, Chem. Phys. 77,435 (1983); 84,463

(1984). ‘H. Partridge, C. W. Bauschlicher, Jr., and P. E. M. Siegbahn, Chem.

Phys. L&t. 97, 198 (1983). 61. Schmidt-Mink, W. Miiller, and W. Meyer, Chem. Phys. 92,263 (1985). ‘E. W. Foster and 0. Richardson, Proc. R. Sot. A 189, 149 (1945). ‘E. W. Foster and 0. Richardson, Proc. R. Sot. A 217,433 (1953). ‘R. A. Frosch and H. M. Foley, Phys. Rev. 88, 1337 (1952). “W. Lichten, Phys. Rev. 120,848 (1960). ’ ‘W. Lichten, Bull. Am. Phys. Sot. Ser. II 7, 43 (1962); 7, 307 (1962);

Phys. Rev. 126, 1020 (1962). r2 W. L. Lichten, Phys. Rev. A 3,594 (1971). I3 P. R. Fontana, Phys. Rev. 125,220 (1962). I4 L. C. Chiu, J. Chem. Phys. 40,2276 (1964). “L. C. Chiu, Phys. Rev. A 137, 384 (1965). 16A. N. Jette and P. Cahill, Phys. Rev. 160, 35 (1967). “A. N. Jette, Phys. Rev. A 5,2009 (1972). I8 A. Carrington, D. H. Levy, and T. A. Miller, Adv. Chem. Phys. 18,149

(1970). “P. R. Brooks, W. Lichten, and R. Reno, Phys. Rev. A 4,2217 (1971). “T. A. Miller and R. S. Freund, J. Chem. Phys. 56,3165 (1972). “T. A. Miller and R. S. Freund, J. Chem. Phys. 58.2345 (1973): “R. S. Freund and T. A. Miller, J. Chem. Phys. 58,3565 (1973). 23 K. F. Freed, J. Chem. Phys. 45,4214 (1966). ‘4 G. York and A. Gallagher, Joint Institute for Laboratory Astrophysics,

University of Colorado, Report No. 114, 1974 (unpublished). =T. Uzer and A. Dalgarno, Chem. Phys. 51,271 (1980). 26D. L. Cooper, J. M. Hutson, and T. Uzer, Chem. Phys. Lett. 86, 472

(1982). ” D. W. Davies and G. J. R. Jones, Chem. Phys. Lett. 81,279 ( 1981). “*G. Baumgartner, H. Kornmeier, and W. Preuss, Chem. Phys. Lett. 107,

13 (1984). “W. Preuss and G. Baumgartner, 2. Phys. A 320, 125 (1985). 3aS. F. Rice, X. Xie, and R. W. Field, Chem. Phys. 104, 161 (1986). “F. Engleke and H. Hage, Chem. Phys. Lett. 103,98 (1983). J2 X. Xie and R. W. Field, Chem. Phys. 99,337 ( 1985). i3 X. ?iie and R. W. Field, J. Mol. Spectrosc. 117,228 (1986). “‘C. Wieman and T. W. Hansch, Phys. Rev. L&t. 36, 1170 (1976). “R. E. Teets, F. V. Kowalski, W. T. Hill, N. Carlson, and T. W. Hans&,

Proc. Sot. Photo-Opt. Instrum. Eng., 113, 80 (1977). “R. Teets, R. Feinberg, T. W. HBnsch, and k L. Schawlow, Phys. Rev.

L&t. 37,683 (1976). ” N. W. Carlson, A. J. Taylor, and A. L. Schawlow, Phys. Rev. Lett. 45,-l 8

(1980). “N W Carlson, A. J. Taylor, K. M. Jones, and A. L. Schawlow, Phys. . .

Rev. A 24,822 (1981); N. W. Carlson, F. V. Kowalski, R. E. Teets, and A. L. Schawlow, Opt. Commun. 29,302 (1979).

I’M. Li, C. Wang, Y. Wang, and L. Li, I. Mol. Spectrosc. 123,161 (1987). @M. Raab, G. Honing, R. Castell, and W. Demtrdder, Chem. Phys. Lett.

66,307 (1979). ” M. Raab, G. Hijning, W. Demtriider, and C. R Vidal, J. Chem. Phys. 76,

4370 (1982). 42 H Gerhardt, T. Huhle, J. Neukammer, and P. J. West, Opt. Commun.

2i, 58 (1978). 4fV. Stert and R. Fischer, Appl. Phys. 17, 151 (1978). 44B. Hemmerling, S. B. Rai, and W. Demtrdder, Z. Phys. A 320, 135

(1985). .“B Hemmerling, R. Bombach, W. Demtriider, and N. Spies, Z. Phys. D 5,

165 (1987). 46B. Hemmerling, R. Bombach, and W. Demtrijder, J. Chem. Phys. 87,

5186 (1987). 47C. R. Vidal and J. Cooper, J. Appl. Phys. 40,337O (1969). 4LI A N Nesmeyanov, Vapor Pressure of the Chemical Elements (Elsevier,

Amsterdam, 1963). p. 127. 49 S. Gerstenkorn, J. Verges, and J. Chevillard, Atlas du Spectre D ‘Absorp-

tion de la Molecule D’lode (CNRS, Paris, 1982).


Jeng et&: Fine structure of Liz 941

“‘H. R. Schlossberg and A. Javan, Phys. Rev. 150,267 (1966). ” M. S. Feld and A. Javan, Phys. Rev. 177,540 (1969). “P. G. Pappas, M. M. Burns, D. D. Hinshelwood, M. S. Feld, and D. E.

Mumick, Phys. Rev. A 21,1955 (1980). “M. Sargent III, Phys. Rev. A 14, 524 (1976). 54W. Demtrijder, LaserSpectroscopy (Springer, New York, 1982), p. 505. ‘s P. H. Vaccaro, Ph.D. thesis, MIT, Cambridge, Massachusetts, 1986. s6M. D. Levenson and G. L. Eesley, Appl. Phys. 19, 1 ( 1979). “R. C. Hilborn, Am. J. Phys. 50,982 (1981). ‘“See AIP document no. PAPS JCPSA-94-928-45 for 45 pages containing

the observed spectral transitions, assignments, differences between observed and calculated frequencies, calculation of the electric dipole matrix elements, and spectral energies, the intensity calculations of the a-pump and ?r-pump spectra, and order by PAPS number and journal reference from American Institute of Physics, Physics Auxiliary Publication Ser- vice, 335 East 45th Street, New York, NY 10017. The price is $1.50 for each microfiche (98 pages) or S5.00 for photocopies of up to 30 pages, and $0.15 for each additional page over 30 pages. Airmail additional. Make check payable to the American Institute of Physics.

s9 H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Di- atomic Molecules (Academic, New York, 1986).

w L. Veseth, Teor. Chim. Acta 18,368 ( 1970).

” J. T. Hougen, The Calculation ofRotational Energy Levelsand Rotatitinal L’ine Intensities in Diatomic Molecules, NBS Monogr. 115 (U.S. GPO, Washington, D.C., 1970).

62A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, Princeton, 1974).

6’ G. Herzberg, Molecular Spectra and Molecular Structure, 2nd ed. (Van Nostrand Reinhold, New York, 1950), Vol. 1.

64Mt E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

65 B. L. Silver, Irreducible Tensor Methods (Academic, New York, 1976). 6bR. N. Zare, Angular Momentum (Wiley, New York, 1988). 6’J. M. Brown and B. J. Howard, Mol. Phys. 31, 1517 (1976). “H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Tko-

Electron Atoms ( Springer-Verlag, Berlin, 1957). 69 L. Li and R. W. Field, J. Mol. Spectrosc. 123,237 (1987). “‘L. Li, Q. Zhu, and R. W. Field, J. Mol. Spectrosc. 134, 50 (1989). ” A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules,

andlons (Springer, Berlin, 1985), p. 101. “J. L. Femtnias, C. Athenour, and R. Stringat, Can. J. Phys. 52, 361

(1974). 73 R. Jost, M. A. Marechal, and M. Lombardi, Phys. Rev. A 5,740 (1972). “‘A. N. Jette and T. A. Miller, Chem. Phys. Lett. 29, 547 (1974).


Fine structure and Zeeman effect in the b(1)3&u state of...

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